In quantum mechanics, the Lévy-Leblond equation describes the dynamics of a spin-1/2 particle.
It was derived by French physicist Jean-Marc Lévy-Leblond in 1967.
[1] Lévy-Leblond equation was obtained under similar heuristic derivations as the Dirac equation, but contrary to the latter, Lévy-Leblond equation is not relativistic.
As both equations recover the electron gyromagnetic ratio, it is suggested that spin is not necessarily a relativistic phenomenon.
For a nonrelativistic spin-1/2 particle of mass m, a representation of the time-independent Lévy-Leblond equation reads:[1] where c is the speed of light, E is the nonrelativistic particle energy,
is the vector of Pauli matrices, which is proportional to the spin operator
By minimal coupling, the equation can be modified to account for the presence of an electromagnetic field,[1] where q is the electric charge of the particle.
This equation is linear in its spatial derivatives.
In 1928, Paul Dirac linearized the relativistic dispersion relation and obtained Dirac equation, described by a bispinor.
This equation can be decoupled into two spinors in the non-relativistic limit, leading to predict the electron magnetic moment with a gyromagnetic ratio
[2] The success of Dirac theory has led to some textbooks to erroneously claim that spin is necessarily a relativistic phenomena.
[3][4] Jean-Marc Lévy-Leblond applied the same technique to the non-relativistic energy relation showing that the same prediction of
[2] Spin is then a result of quantum mechanics and linearization of the equations but not necessarily a relativistic effect.
[3][5] Lévy-Leblond equation is Galilean invariant.
This equation demonstrates that one does not need the full Poincaré group to explain the spin 1/2.
, quantum mechanics under the Galilean transformation group are enough.
[1] Similarly, one can construct a non-relativistic linear equation for any arbitrary spin.
[1][6] Under the same idea one can construct equations for Galilean electromagnetism.
[1] Taking the second line of Lévy-Leblond equation and inserting it back into the first line, one obtains through the algebra of the Pauli matrices, that[3] which is the Schrödinger equation for a two-valued spinor.
also returns another Schrödinger's equation.
Pauli's expression for spin-1⁄2 particle in an electromagnetic field can be recovered by minimal coupling:[3] While Lévy-Leblond is linear in its derivatives, Pauli's and Schrödinger's equations are quadratic in the spatial derivatives.
is the total relativistic energy.
one recovers, Lévy-Leblond equations.
Similar to the historical derivation of Dirac equation by Paul Dirac, one can try to linearize the non-relativistic dispersion relation
, such that their product recovers the classical dispersion relation, that is where the factor 2mc2 is arbitrary an it is just there for normalization.
By carrying out the product, one find that there is no solution if
are matrices that must satisfy the following relations: these relations can be rearranged to involve the gamma matrices from Clifford algebra.
is the Identity matrix of dimension N. One possible representation is such that
, returns Lévy-Leblond equation.
Other representations can be chosen leading to equivalent equations with different signs or phases.